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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#relation_to_tqft'>Relation to TQFT</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#expositions'>Expositions</a></li> <li><a href='#via_geometric_quantization'>Via geometric quantization</a></li> <li><a href='#as_a_tcft'>As a TCFT</a></li> <li><a href='#general'>General</a></li> <li><a href='#in_higher_differential_geometry__on_orbifolds'>In higher differential geometry / on orbifolds</a></li> <li><a href='#in_terms_of_motives'>In terms of motives</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Gromov-Witten theory</em> studies <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a> via maps of <a class="existingWikiWord" href="/nlab/show/Riemann+surfaces">Riemann surfaces</a> into it. In contrast to the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, where one studies maps from the circle into the space, there is no composition law for Riemann surfaces. Instead, one considers <a class="existingWikiWord" href="/nlab/show/pseudoholomorphic+curves">pseudoholomorphic curves</a> in a fixed homology class and with fixed boundary conditions such that only a finite number of such maps exists. These numbers are symplectic invariants. Their computation is difficult and uses enumerative and analytic techniques from <a class="existingWikiWord" href="/nlab/show/Floer+homology">Floer homology</a>.</p> <p>Gromov-Witten invariants are used to deform the cup product of cohomology classes (using only maps from the Riemann sphere into the manifold), leading to <a class="existingWikiWord" href="/nlab/show/quantum+cohomology">quantum cohomology</a>.</p> <p>Gromov-Witten theory originates in physics from the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>. A powerful tool to compute Gromov-Witten invariants is <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a>.</p> <p>The precise mathematical definition uses the notion of the moduli space of <a class="existingWikiWord" href="/nlab/show/stable+maps">stable maps</a>.</p> <h2 id="definition">Definition</h2> <p>…</p> <h2 id="relation_to_tqft">Relation to TQFT</h2> <p>Gromov-Witten invariants may be understood (and have originally been found) as arising from a particular <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>, or actually a <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>, called the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>.</p> <p>For a useful exposition of this see (<a href="#Tolland">Tolland</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+map">stable map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+sheaf+cohomology">quantum sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Donaldson+theory">Donaldson theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GW%2FDT+correspondence">GW/DT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enumerative+geometry">enumerative geometry</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="expositions">Expositions</h3> <p>Introductory notes:</p> <ul> <li> <p>Simon Rose, <em>Introduction to Gromov-Witten theory</em> (<a href="http://arxiv.org/abs/1407.1260">arXiv:1407.1260</a>)</p> </li> <li id="Bertram02"> <p><a class="existingWikiWord" href="/nlab/show/Albrecht+Bertram">Albrecht Bertram</a>, <em>Stable Maps and Gromov-Witten Invariants</em>, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 (<a href="http://users.ictp.it/~pub_off/lectures/lns019/Bertram/Bertram.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Bertram_StableMaps.pdf" title="pdf">pdf</a>)</p> </li> <li id="Katz06"> <p><a class="existingWikiWord" href="/nlab/show/Sheldon+Katz">Sheldon Katz</a>, <em>Enumerative Geometry and String Theory</em>, Student Mathematical Library <strong>32</strong>, AMS 2006 (<a href="https://bookstore.ams.org/stml-32">ISBN:978-1-4704-2143-4</a>, <a href="https://inspirehep.net/literature/739788">spire:739788</a>)</p> </li> </ul> <p>Seminar notes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+ideas+of+moduli+stacks+of+curves+and+Gromov-Witten+theory">basic ideas of moduli stacks of curves and Gromov-Witten theory</a></li> </ul> <p>And this introductory bit on the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a>s:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology+-+elliptic+curves">A Survey of Elliptic Cohomology - elliptic curves</a>.</li> </ul> <p>An exposition of GW theory as a <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> is at</p> <ul> <li id="Tolland"><a class="existingWikiWord" href="/nlab/show/AJ+Tolland">AJ Tolland</a>, <em>Gromov-Witten Invariants and Topological Field Theory</em> (<a href="http://sbseminar.wordpress.com/2008/12/11/gromov-witten-invariants-and-topological-field-theory">blog</a>)</li> </ul> <p>The origin of Gromov-Witten theory in and relation to <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> and other <a class="existingWikiWord" href="/nlab/show/physics">physics</a> motivation is recalled and surveyed in</p> <ul> <li>Daniel Grunberg, <em>Gromov-Witten Theory and Threshold Corrections</em> (<a href="http://arxiv.org/abs/hep-th/0605087">arXiv:hep-th/0605087</a>)</li> </ul> <h3 id="via_geometric_quantization">Via geometric quantization</h3> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> is in</p> <ul> <li id="CladerPriddisShoemaker13">Emily Clader, Nathan Priddis, Mark Shoemaker, <em>Geometric Quantization with Applications to Gromov-Witten Theory</em> (<a href="http://arxiv.org/abs/1309.1150">arXiv:1309.1150</a>)</li> </ul> <h3 id="as_a_tcft">As a TCFT</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>The Gromov-Witten potential associated to a TCFT</em> (<a href="http://arxiv.org/abs/math/0509264">arXiv:0509264</a>)</li> </ul> <p>See also the references at <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>.</p> <h3 id="general">General</h3> <p>A discussion by quantization of quadratic Hamiltonians is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Givental">Alexander Givental</a>, <em>Gromov-Witten invariants and quantization of quadratic Hamiltonians</em> (<a href="http://math.berkeley.edu/~giventh/papers/gwi.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Gromov-Witten classes, quantum cohomology, and enumerative geometry</em>, Comm. Math. Phys. 164 (1994), no. 3, 525–562 (<a href="http://projecteuclid.org/euclid.cmp/1104270948">euclid</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Frobenius manifolds, quantum cohomology and moduli spaces</em>, Amer. Math. Soc., Providence, RI, 1999,</p> </li> <li> <p>W. Fulton, R. Pandharipande, <em>Notes on stable maps and quantum cohomology</em>, in: Algebraic Geometry, Santa Cruz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45–96 (1997)</p> </li> <li> <p>J Robbin, D A Salamon, <em>A construction of the Deligne-Mumford orbifold</em>, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611–699 (<a href="http://arxiv.org/abs/math/0407090">arxiv</a>; <a href="http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=8&iss=4&rank=3">pdf at JEMS</a>); corrigendum J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 901–905 (<a href="http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=9&iss=4&rank=11">pdf at JEMS</a>).</p> </li> <li> <p>J Robbin, Y Ruan, D A Salamon, <em>The moduli space of regular stable maps</em>, Math. Z. 259 (2008), no. 3, 525–574 (<a href="http://dx.doi.org/10.1007/s00209-007-0237-x">doi</a>).</p> </li> <li> <p>Martin A. Guest, <em>From quantum cohomology to integrable systems</em>, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.</p> </li> <li> <p>Joachim Kock, Israel Vainsencher, <em>An invitation to quantum cohomology. Kontsevich’s formula for rational plane curves</em>, Progress in Mathematics, 249. Birkhäuser Boston, Inc., Boston, MA, 2007. xiv+159 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, <a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar Salamon</a>, <em>Introduction to symplectic topology</em>, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.</p> </li> <li> <p>Sheldon Katz, <em>Enumerative geometry and string theory</em>, Student Math. Library <strong>32</strong>. IAS/Park City AMS & IAS 2006. xiv+206 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eleny-Nicoleta+Ionel">Eleny-Nicoleta Ionel</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+H.+Parker">Thomas H. Parker</a>, <em>Relative Gromov-Witten invariants</em>, Ann. of Math. (2) 157 (2003), no. 1, 45–96 (<a href="http://dx.doi.org/10.4007/annals.2003.157.45">doi</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/AJ+Tolland">AJ Tolland</a>, <em>Gromov-Witten Gauge Theory I</em> (<a href="http://arxiv.org/abs/0904.4834">arXiv:0904.4834</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>The structure of 2D semi-simple field theories</em> (<a href="http://arxiv.org/abs/0712.0160">arXiv:0712.0160</a>)</p> </li> <li> <p>Oliver Fabert, <em>Floer theory, Frobenius manifolds and integrable systems</em>, (<a href="http://arxiv.org/abs/1206.1564">arxiv/1206.1564</a>)</p> </li> </ul> <p>A generalization is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, A. Losev, <a class="existingWikiWord" href="/nlab/show/Nikita+Nekrasov">Nikita Nekrasov</a>, <em>Instantons beyond topological theory I</em> (<a href="http://arxiv.org/abs/hep-th/0610149">arXiv:hep-th/0610149</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, A. Losev, <a class="existingWikiWord" href="/nlab/show/Nikita+Nekrasov">Nikita Nekrasov</a>, <em>Instantons beyond topological theory II</em> (<a href="http://arxiv.org/abs/0803.3302">arXiv:hep-th/0610149</a>)</p> </li> </ul> <p>Expositions and summaries of this are in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, A. Losev, <a class="existingWikiWord" href="/nlab/show/Nikita+Nekrasov">Nikita Nekrasov</a>, <em>Notes on instantons in topological field theory and beyond</em> (<a href="http://arxiv.org/abs/hep-th/0702137">arXiv:hep-th/0702137</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <em>Localized</em> (2006) (<a href="Localized">blog</a>)</p> </li> </ul> <h3 id="in_higher_differential_geometry__on_orbifolds">In higher differential geometry / on orbifolds</h3> <p>GW theory of <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> (hence in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>) has been introduced in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Weimin+Chen">Weimin Chen</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, <em>Orbifold Gromov-Witten Theory</em>, in <em>Orbifolds in mathematics and physics</em> (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (<a href="http://arxiv.org/abs/math/0103156">arXiv:math/0103156</a>)</li> </ul> <p>A review with further pointers is in</p> <ul> <li>Dan Abramovich, <em>Lectures on Gromov-Witten invariants of orbifolds</em> (<a href="http://arxiv.org/abs/math/0512372">arXiv:math/0512372</a>)</li> </ul> <h3 id="in_terms_of_motives">In terms of motives</h3> <p>That the <a class="existingWikiWord" href="/nlab/show/path+integral+as+a+pull-push+transform">pull-push quantization</a> of Gromov-Witten theory is naturally understood as a “<a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a>” in terms of <a class="existingWikiWord" href="/nlab/show/Chow+motives">Chow motives</a> of <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stacks">Deligne-Mumford stacks</a> was suggested in</p> <ul id="BehrendManin95"> <li><a class="existingWikiWord" href="/nlab/show/Kai+Behrend">Kai Behrend</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Stacks of Stable Maps and Gromov-Witten Invariants</em> (<a href="http://arxiv.org/abs/alg-geom/9506023">arXiv:alg-geom/9506023</a>)</li> </ul> <p>Further investigation of these stacky Chow motives then appears in</p> <ul id="Toen00"> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <em>On motives for Deligne-Mumford stacks</em>, International Mathematics Research Notices 2000, 17 (2000) 909-928 (<a href="http://arxiv.org/abs/math/0006160">arXiv:math/0006160</a>, <a href="http://hal.archives-ouvertes.fr/hal-00773027">web</a>, <a href="http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf">pdf</a>)</li> </ul> <ul> <li>Utsav Choudhury, <em>Motives of Deligne-Mumford Stacks</em> (<a href="http://arxiv.org/abs/1109.5288">arXiv:1109.5288</a>)</li> </ul> <p>On its relation to <a class="existingWikiWord" href="/nlab/show/tropical+geometry">tropical geometry</a>:</p> <ul> <li> <p><span class="newWikiWord">Grigory Mikhalkin<a href="/nlab/new/Grigory+Mikhalkin">?</a></span>. <em>Enumerative Tropical Algebraic Geometry in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math></em>. Journal of the American Mathematical Society, Vol. 18, No. 2 (Apr., 2005), pp. 313-377. (<a href="https://doi.org/10.1090/S0894-0347-05-00477-7">doi</a>)</p> </li> <li> <p>Emil Albrychiewicz, Kai-Isaak Ellers, Andrés Franco Valiente, <a class="existingWikiWord" href="/nlab/show/Petr+Ho%C5%99ava">Petr Hořava</a>. <em>Tropological Sigma Models</em>. (2023). (<a href="https://arxiv.org/abs/2311.00745">arXiv:2311.00745</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 7, 2024 at 23:52:43. See the <a href="/nlab/history/Gromov-Witten+invariants" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Gromov-Witten+invariants" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/13271/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/Gromov-Witten+invariants/33" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Gromov-Witten+invariants" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Gromov-Witten+invariants" accesskey="S" class="navlink" id="history" rel="nofollow">History (33 revisions)</a> <a href="/nlab/show/Gromov-Witten+invariants/cite" style="color: black">Cite</a> <a href="/nlab/print/Gromov-Witten+invariants" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Gromov-Witten+invariants" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>